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%% hw6.tex
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%% Started on  Wed Oct  5 12:32:20 2011 alex
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\section*{Homework 6}\renewcommand{\leftmark}{Homework 6}\phantomsection\addcontentsline{toc}{section}{Homework 6}

\begin{exercise}
Let
\begin{equation}
f(t)=\begin{cases}
0, &\mbox{ if }t\leq0,\\
\lfloor t\rfloor+1, &\mbox{ if }t>0
\end{cases}
\end{equation}
(where $\lfloor{t}\rfloor$ denotes the maximal integer in the interval $(-\infty, t]$). Find (an explicit formula for) the Laplace transform $\widetilde{f}(z)$. What is $\sigma$?
\end{exercise}
\begin{exercise}
The same for the ``saw-function''
\begin{equation}
f(t)=\begin{cases}
0 & \mbox{if }t\leq0\\
t-2n & \mbox{if }2n\leq t<2n+1\\
2n+2-t & \mbox{if }2n+1\leq t<2n+2
\end{cases}
\end{equation}
\end{exercise}
\begin{exercise}
Let
\begin{equation}
f(x)=\begin{cases}0,&\mbox{if }x<0,\\
1,&\mbox{if }x\geq0
\end{cases},\quad
g(x)=\begin{cases} 0, &\mbox{ if }x<0\\
x,&\mbox{if }x\geq0.
\end{cases}
\end{equation}


Find $f*f$, $f*g$, $g*g$.
\end{exercise}
\begin{exercise}
Under the appropriate assumptions (you need to formulate them),
prove the identities $f*g=g*f, (f*g)'=f*g'=f'*g$. 
\end{exercise}
